18 problems

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Schubert problem for $\mathbb{F}\ell(\alpha;n)$. For each permutation $w_i$, let…

Let $\rho_n:S^1\to\mathbb{R}P^n$ be the real rational normal curve, and let $\sharp_{k,n}$ denote the degree of the Grassmannian of projective $k$-dimensional subspaces in…

Let $X$ be an algebraic set defined over $\mathbb{R}$. It has the M-property when the sum of the $\mathbb{Z}/2\mathbb{Z}$-Betti numbers of $X(\mathbb{R})$ equals the corresponding…

Let $\gamma$ be the rational normal curve, and for a set $S_i=\{s_{i1},\dotsc,s_{in}\}$ of $n$ distinct points of $\mathbb{R}\mathbb{P}^1$, let $F_\bullet(S_i)$ be the flag whose…

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $\mathbb{F}\ell(\alpha;n)$ be the corresponding flag manifold. Let $(w_1,\dotsc,w_m)$ be a Schubert problem, an…

Let $W_1,\ldots,W_q$ be projective subspaces in general position in $\mathbf{P}^d$, with $a_j={\mathrm{dim}\,} W_j$ satisfying … Let $E:\mathbf{P}^1\to\mathbf{P}^d$ be the rational…

Let $C_d\subseteq\mathbb P^d$ be the rational normal curve of degree $d\geq2$, and let its RR discriminant be decomposed into nonisotropic and isotropic parts. Rational normal curv…

Let $\widetilde{\mathcal{C}}$ be the rational normal curve and let $p\in\operatorname{\mathbb{P}}^{r+1}\setminus\widetilde{\mathcal{C}}^3$ be a projection center, with…

Let $C\subseteq \mathbf{P}^d$ be a rational normal curve of degree $d$, and let … be its $k$-th secant variety, defined as the closure of the union of $(k+1)$-secant $k$-planes to…

Let $C\subset\mathbb{P}^n$ be a rational normal curve, and let $X_n=\operatorname{Bl}_C\mathbb{P}^n$. For an integer $d$, write $\operatorname{Sec}_d(C)$ for the $d$-secant variety…

Conjecture on special effect varieties for $n+3$ points. In this situation, the special effect varieties are linear spans of points, secant varieties of the rational normal curve,…

Let ${\mathcal L}$ be a non-empty linear system in ${\mathbb P}^n$ with $n+3$ base points in general position, and let $C$ be the rational normal curve through those points. Let…

Let $B$ be the homogeneous coordinate ring of a rational normal curve, and let $\mathbb{B}(B)$ denote the cone of Betti tables of finitely generated $B$-modules. For each $c=0,1,2$…

Monotone Secant Conjecture. For any Grassmannian Schubert problem $(\sigma_1,\ldots, \sigma_m)$ on $\mathbb{F}\ell(\alpha;n)$ and any disjoint secant flags…

Let $G(k,n)$ be a Grassmannian, let $\gamma$ be a rational normal curve, and let $\lambda^1,\dotsc,\lambda^m$ be a Schubert problem on $G(k,n)$. A generalized secant subspace to…

For each positive integer $d$, let $(\alpha_1,\ldots,\alpha_{e_d-1})$ be the saturation sequence of the rational normal curve of degree $d$. Terminal-3 conjecture. For every positi…

For each integer $d\geqslant6$, let $\alpha_1$ and $\alpha_2$ denote the first two entries of the saturation sequence of the rational normal curve of degree $d$. Strict inequality…

Monotonicity conjecture. The saturation sequence is non-increasing: